Improved explicit estimates for the discrete Laplace operator with hyperbolic circle patterns

TL;DR

This paper provides explicit estimates for the discrete Laplace operator in hyperbolic circle patterns, aiding in proving long-term existence of combinatorial Calabi flows in hyperbolic geometry.

Contribution

It introduces two explicit estimates for the discrete Laplace operator in hyperbolic geometry, improving understanding of flow solutions in this setting.

Findings

Established new explicit estimates for the discrete Laplace operator

Provided alternative proofs for long-term existence of combinatorial Calabi flows

Enhanced analytical tools for hyperbolic circle pattern analysis

Abstract

Ge in his thesis \cite{Ge-thesis} introduced the combinatorial Calabi flows and established the long time existence and convergence of solutions to the flows in both hyperbolic and Euclidean background geometries. It is noteworthy that the existence of solutions to the combinatorial Calabi flows in hyperbolic background geometry proves to be more intricate and challenging compared to the Euclidean background geometry. The main difficulty is to establish the compactness, especially the lower boundeness along the flow equations. In this paper, we give two explicit estimates for the discrete Laplace operator based on the Glickenstein-Thomas formulation \cite{Glickenstein2017} for discrete hyperbolic conformal structures. As applications, we give new proofs of the long time existence of solutions to the combinatorial Calabi flows established by Ge-Xu \cite{Ge2016}, Ge-Hua \cite{Ge2018} and the combinatorial $p$-th Calabi flows established by Lin-Zhang \cite{Lin2019} in hyperbolic background geometry.